Limits and Continuity Index

Continuity Theorems.

The basic theorems based on continuity are given below: If the functions f(x) and g(x) are continuous at x=a then, 1> is continuous at x=a. 2> is continuous at x=a. 3> is continuous at x=a , if g(x) is not equal to 0. 4> is continuous at x=a if f(x) is greater than 0 , or is a positive number when “n” is...

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Continuity of a function(continuous and discontinuous functions).

Continuity of a function(continuous and discontinuous functions).

A function “f” in interval [a,b] is said to be a continuous function when the Graph drawn for f(x) is a smooth line or curve without any break in it. Such curve or line can be drawn by the continuous motion of a pencil in a sheet of paper. And Discontinuous  function is just opposite of the continuous function , the function “f” is said to be  discontinuous function when the graph drawn for f(x) is  consists of disconnected curves or lines. For example: Continuous Function: Discontinuous Function: If we zoom into...

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Basic properties or theorems of limit.

The limit theorems or basic properties of limit are given below: 1> The limit of the sum (or difference) of the functions “f” and “g” is the sum (or difference) of the limits of the functions i.e. 2> The limit of the product of the function “f” and “g” is the product of the limits of the functions. i.e. 3> The limit of the quotient of the function “f” and “g” is the quotient of the limits of the functions. i.e. 4> The limit of nth root of a function...

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Right hand and Left hand limit of a function.

Right hand and Left hand limit of a function.

Let an Interval be denoted by (a-β , a+β) which is shown by the figure below: and x∈ (a-β , a+β) And let a function f(x) be defined at the Interval (a-β , a+β) . Then we can also find the limit of  function f(x) as, Left hand Limit of a Function: In the above  case the limit of f(x) when “x” approaches “a” from the left hand side of the interval is known as the left hand limit of f(x). and is denoted by: Right hand Limit of a Function: Similarly, the limit of f(x) when...

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Interval.

What is Interval? A set of points lying between any two points “a” and “b” is known as an Interval. For example: the interval from points x=1 to x=10 is the set of points lying between 1 and 10. Open and Closed Interval: An Interval which includes it’s end points(a and b)  is known as Closed interval ; While an Interval Which does not includes it’s end points(a and b)  is known as Open Interval. A open Interval from point “a” to point “b” is denoted by: (a,b) and  A closed...

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The concept of Limit.

The concept of Limit.

What is limit? First study the following picture: You can see in the figure ; if a polygon is inscribed inside a circle , then as we increase the number of sides of polygon the area of polygon gradually increases . As we increase the number of sides in the polygon it’s area gradually approaches the area of  circle and if we take a polygon with sufficiently large number of sides in same configuration then , the area of the polygon will become almost equal to the area of circle  , but it will never be equal to   the area of the...

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